Hypercyclic Abelian Semigroups of Matrices on $\mathbb{R}^n$

TL;DR

This paper investigates the minimal number of matrices needed to generate hypercyclic abelian semigroups on real n-dimensional space, establishing bounds and correcting previous results in the literature.

Contribution

It determines the minimal number of matrices in normal form over  that form a hypercyclic abelian semigroup on , correcting earlier published results.

Findings

No abelian semigroup generated by  matrices can be hypercyclic on 

The minimal number of generators for hypercyclicity is  or more

The paper corrects and clarifies previous results in the literature

Abstract

In this paper, we bring together results about the existence of a somewhere dense (resp. dense) orbit and the minimal number of generators for abelian semigroups of matrices on $\mathbb{R}^n$. We solve the problem of determining the minimal number of matrices in normal form over $\mathbb{R}$ which form a hypercyclic abelian semigroup on R^n. In particular, we show that no abelian semigroup generated by $[\frac{n+1}{2}]$ matrices on $\mathbb{R}^n$ can be hypercyclic. ([ ] denotes the integer part). This is a corrected version of the paper published in Topology and its Applications 210 (2016), 29-45 (see also [4]). The differences between this version and the published version are explained at the end of the Introduction.